Orbital magnetization in Sierpinski fractals

Orbital magnetization (OM) in Sierpinski carpet (SC) and triangle (ST) fractal is theoretically investigated by using Haldane model as a prototypical example. The OM calculation is performed following two distinct approaches; employing the definition and local markers formalism. Both methods coincides for all systems analyzed. For the SC, higher fractal generations create a dense set of edge states, resulting in a staircase profile, leading to oscillations in the magnetization as a function of the chemical potential. In contrast, the ST self-similarity produces distinct fractal-induced spectral gaps, which manifest as constant plateaus in the magnetization. The STs exhibit a pronounced sensitivity to edge terminations. Our results reveal how quantum confinement in fractal structures affects the electronic orbital angular momentum, pointing to possible pathways for exploring novel orbitronics in systems with complex geometries.